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two identical spheres are released from a device at time equals zero from the same height H as shown above or T equals zero I should say CRA has no initial velocity and falls straight down spear B is given an initial horizontal velocity of magnitude V Sub Zero and travels a horizontal distance D before it reaches the ground the spheres reach the ground at the same time T sub F even though sphere B has more distance to cover before landing air resistance is negligible the dots below represent spheres a and B draw a Freebody diagram showing and labeling the forces not components exerted on each sphere at time T sub F over two so we can see our spheres here when I guess this little this thing releases for your a goes straight down it's fear B it it will go well it's vertical in the vertical direction it'll go down just the same way it'll be accelerated in just the same way sphere a but it has some horizontal velocity that makes it move out and hit the ground D to the right and when it hits the ground that's T sub F when they're up here that's right when they're released as T equals zero and then this is at T equals T sub F and they say a Freebody diagram at T sub F over two so this is y while both of them are in flight so well both of them are in flight the only force acting on each of them is just going to be the force of gravity and since the spheres are identical the force of gravity is going to be identical they have the same mass so let me draw so that right over there is the force of gravity on the sphere a and that is the force of gravity on sphere B and so we could write force of gravity force of force of gravity and if we want we could call well we could say the magnitude is F sub G if we want F sub G or we could label it as M times the gravitational field so this is to is equal to M times the gravitational field and that's it then while they're mid-flight the only force acting on them we're assuming air resistance is negligible is the force of gravity is going to be the same because they have the same mass they're identical spheres all right let's tackle the next part of this on the axes below sketch and label a graph of the horizontal components of the velocity of sphere a and a sphere B as a function of time all right I'll do sphere a first this is pretty straightforward C or a if you will remember let's go up here skier a has no horizontal velocity the entire time we're talking about it only it's only going to be accelerated in the vertical direction it's going to be accelerated downwards so sphere a has no horizontal velocity so I will draw a line like this so sphere a has no horizontal velocity the entire time now sphere sphere B sphere B is going to be a little bit more interesting slightly more interesting it's velocity they tell us that its initial velocity is V sub 0 its initial horizontal velocity I should say has a magnitude of V sub 0 and since air resistance is negligible it's going to continue going to the right at V sub 0 until it hits the ground so so sphere B if this is and I'm just going to pick one of these as V sub 0 let's say that this right over here is V sub 0 that's the magnitude of its horizontal velocity well sphere B is going to be at that velocity actually let me just make it a little bit clearer it's going to be at that velocity until until V F so if we say this right over here are not VF until the final time until T F so this is T equals 0 TF that entire time while the balls in the well that sphere is in the air it's going to have a the horizontal component of its velocity is just going to be constant it's not going to be slowed down by anything because we're assuming air resistance is negligible and then right when it hits the ground and essentially if you think about the force that is stopping it it's essentially friction but then it very quickly goes down to a velocity of a magnitude of velocity of horizontal magnitude of velocity of zero all right all right now let's tackle the last part of this now you could label this if you want this is let me actually let me label it this is B C or B and this is sphere that is a sphere a right over there and sphere B if you want you could show it would overwrite sphere AC or B would be 0 after that it's not continuing to move on to the right or at least they don't tell us anything about about that finally in a clear coherent paragraph in a clear coherent paragraph length responds explain why the spheres reach the ground at the same time even though they travel different distances include references in your answers to Parts A and B all right so let me think about it I'll try to write a clear coherent paragraph length response so I'll say the entire time the or let me say from from T equals 0 to T equals T sub F the only force acting on the spheres is the downward force of gravity is the downward force force of write a little bit neater of gravity at T equals 0 at T equals 0 they both they both have zero vertical velocity or the magnitude of the velocity in the vertical direction is zero for both of them let me write it that way the the magnitude of both of their velocities both of their velocities velocities in the vertical direction is zero after T equals zero they are accelerated they are accelerated at the same rate accelerated at the same they are accelerated at the same rate so their vertical component of velocity their vertical component components of velocity velocity are always the same of velocity are always the same and they have the same vertical distance to cover and they have the same the same vertical distance to cover equal distance to cover so they hit the ground at the same time let me make sure that makes sense after T equals zero they are accelerated at the same rate so their vertical components of velocity are always the same let me actually let me let me write this this way since they have the same since let me since they have the same vertical distance to cover vertical distance to cover they will hit the ground at the same time they will hit the ground at the same time same time they do have different horizontal velocities but that does not affect their that does not affect the time their velocities or the distance in the vertical direction they have different horizontal horizontal velocities but that does not affect the time in which they cover the same vertical distance affect the time in which they cover the same vertical distance and you could write something to that effect and you could also write that yes if you were to add the components of spheres bees velocities it would actually have a larger velocity if you were to add the components if you're not thinking you needed the horizontal or the vertical direction and so it does indeed cover more distance in space over the same amount of time but if you think about it just in the vertical direction it's covering the same distance in the same time at any given point in time in the vertical direction it actually has the same velocity it's being accelerated in the same way and it starts off at the loss of magnitude of velocity of zero

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